From Fractal Urban Pattern Analysis to Fractal Urban Planning Concepts - Computational Approaches for Urban Environments - page 16

an increasing number of smaller and smaller square-like elements. Indeed at each
step, the number of elements is multiplied by N and their size is reduced by r so
that at the n th step, we have N n elements of length r n L ,where L is the length of
the initial square. So, both the number of elements and their size follow geometric
series. Moreover, at each step, an increasing number of smaller and smaller lacunae
are generated, and hence, a hierarchical system of lacunae appears (Fig. 2.1 d).
This is one of the most striking features of these fractals. At the same time, the
boundary line of the Sierpinski carpet is lengthened, since a growing number of
smaller indents are generated. Hence, the boundary length tends to infinity, whereas
the surface area tends to zero, and the limit geometrical object is neither two
dimensional like a surface nor one-dimensional like a line. This is one of the
most striking features of such fractals. Euclidian objects like squares or circles are
compact and their border is smooth—except at some singular points, the corners—
what never can be for fractals.
This led mathematicians to introduce the notion of fractal dimension in order
to characterize these objects which are made up of elements that are distributed
highly nonuniformly in space. The basic requirement is that there exists a measure M
which remains constant throughout the iteration and is characteristic of the fractal.
This measure is defined by computing the total “mass” of the fractal at each step,
which is the product of the number of elements and their size weighted by the free
parameter D , the fractal dimension

.r n L/ D N n L D D M (2.1)

This relation yields

log N
log r

N n D .r n / D ) D D (2.2)

so that D is indeed a constant over all iterations and, in the case of constructed
fractals, is related to the two parameters N and r. Hence, for the Sierpinski carpet
of Fig. 2.3 , we obtain D D 1.47 (Mandelbrot 1982 ; Frankhauser 1994 ). Obviously,
the dimension does not depend on the position of the elements within the square
within which they were generated. Hence, we may randomly change their position
without affecting fractal properties, and accordingly, irregular empirical structures
may display fractal properties. However, fractality would be disturbed by putting
squares in lacunae generated in previous iterations since this would affect the lacunal
hierarchy. For the same reason, squares are not allowed to intersect.
It is also possible to calculate the fractal dimension of the edge of the fractals.
For Sierpinski carpets, the dimension of boundaries is equal to that of the surface
area. This may come as a surprise, but it reflects the fact that the boundary, like the
surface area, converges to the same limit set. This fits in with the observation that,
for metropolitan areas, the length of the outline of the clusters is proportional to the
built-up area. In fractal terms, this would mean that both the edge and the surface
have the same fractal dimension, like a Sierpinski carpet.